Strong underrelaxation in Kaczmarz's method for inconsistent systems

Yair Censor, Paul P.B. Eggermont, Dan Gordon

Research output: Contribution to journalArticlepeer-review


We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.

Original languageEnglish
Pages (from-to)83-92
Number of pages10
JournalNumerische Mathematik
Issue number1
StatePublished - Feb 1983


  • Subject Classifications: AMS (MOS): 65F10, CR: 5.14

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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